Two-level Lambda-calculus

Two-level lambda-calculus is designed to provide a mathematical model of capturing substitution, also called instantiation. Instantiation is a feature of the ‘informal meta-level’; it appears pervasively in specifications of the syntax and semantics of formal languages. The two-level lambda-calculus has two levels of variable. Lambda-abstraction and beta-reduction exist for both levels. A level 2 beta-reduct, triggering a substitution of a term for a level 2 variable, does not avoid capture for level 1 abstractions. This models meta-variables and instantiation as appears at the informal meta-level. In this paper we lay down the syntax of the two-level lambda-calculus; we develop theories of freshness, alpha-equivalence, and beta-reduction; and we prove confluence. In doing this we give nominal terms unknowns — which are level 2 variables and appear in several previous pa-pers — a functional meaning. In doing this we take a step towards longer-term goals of developing a foundation for theorem-provers which directly support reasoning in the style of nominal rewriting and nominal algebra, and towards a mathematics of functions which can bind names in their arguments